Regarding nonzero roots, it was shown in [1]} that for the original Yablonskii-Vorob'ev polynomials \(Q_{N}(z)\) , all nonzero roots are simple. For the higher Yablonskii-Vorob'ev polynomial hierarchy \(Q_{N}^{[m]}(z)\) , it was conjectured in [2]} that all nonzero roots are also simple.
In view of Theorem 2, this conjecture implies that the polynomial \(Q_{N}^{[m]}(z)\)  has
\( N_p=\frac{1}{2}\left[N(N+1)-N_{0}(N_{0}+1)\right]\) 
